202 



Mr. Harley also laid before the Society the following 

 communication from Mr. Cockle, dated Temple, February, 

 1862. 



The theorem which I gave in my first paper, '' On Tran- 

 scendental and Algebraic Solution," in the Philosophical 

 Magazine for May, 1861, viz., that the solution of an 

 algebraic equation of the n-th degree, whereof the coefficients 

 are functions of one parameter, depends upon that of a linear 

 differential equation of the (n — 1) th order, may be applied 

 to any algebraic equation whatever, without any preliminary 

 modification of its coefiicients. And the simplest, or nearly 

 the simplest, form of the general process indicated in my 

 second paper (^' Supplementary Paper," Philoso2yhical Maga- 

 zine for February, 1862) wdth the above title is, as I have 

 pointed out to Mr. Harley, obtained by treating all the 

 coefficients as constant and multiplying the last into a para- 

 meter X, which is to be treated as the independent variable. 

 Thus, given the quadratic 



fJ^hy^cz=:0, 

 we deduce, successively, from 



the following relations :— 



In this expression x is to be made equal to unity, and the 

 arbitrary constant C determined by substituting for y in the 

 given quadratic, or by processes which will, I hope, soon be 

 explained by Mr. Harley. For the general sextic I should 

 be disposed to deal with the form (attainable by Mr. Jerrard's 

 process, by vanishing groups, or by Mr. Sylvester's method)^ 



